Perron–Frobenius theorem

In matrix theory, the Perron–Frobenius theorem, proved in its first part by Oskar Perron (1907) and extended by Georg Frobenius (1912), asserts that a real square matrix with positive entries has a unique eigenvalue of largest magnitude and that eigenvalue is real. The corresponding eigenvector can be chosen to have strictly positive components, and also asserts a similar statement for certain classes of nonnegative matrices. This theorem has important applications to probability theory (ergodicity of Markov chains); to the theory of dynamical systems (subshifts of finite type); to economics (Okishio's theorem, Hawkins–Simon condition);

Perron–Frobenius theorem - Wikipedia Jump to content From Wikipedia, the free encyclopedia Theorem in linear algebra In matrix theory , the Perron–Frobenius theorem , proved in its first part by Oskar Perron  ( 1907 ) and extended by Georg Frobenius  ( 1912 ), asserts that a real square matrix with positive entries has a unique eigenvalue of largest magnitude and that eigenvalue is real. The corresponding eigenvector can be chosen to have strictly positive components, and also asserts a similar statement for certain classes of nonnegative matrices . This theorem has important

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